predict customer visit
What interests us in this problem are only the intervals for 1 person. Lets say that we want to train a neural network on recognizing the simple pattern in date differences. This would mean that we could train the neural network on series of purchase histories of multiple people. That means that one possible input is the previous intervals: in your case the previous 2 intervals (10 days, 15 days). That is a very small number and it will be hard to recognise a pattern for 3 consecutive purchases, but lets disregard that. After all, you can always experiment with different input sizes. 

For training, you can take multiple customers, where of all of them you know when 4 purchases took place. You calculate the intervals, thus for each customer you get 3 intervals. As input you give the network 2, you let the network predict the 3rd one, and then you let the network learn off its mistake (you can do that because you know what the 3rd one was in reality, that's how you train the network).

Provided that the network isn't too complex or too simple (by that I mean that a hidden layer could be necessary, and you need to give it sufficient neurons) than the network might give a decent approximation. The only problem is, as Manuel Rodriguez said, that the network would indeed only learn to make predictions based on the corectness of its previous ones on a large group of people. Perfect predictions thus won't be possible, but you can minimize the error by making your dataset and input larger, so the network has more information to work on.